Lecture 9: Kinetic Energy and Work 1 CHAPTER 6: Work and Kinetic Energy The concept of WORK has a very precise definition in physics. Work is a physical quantity produced when a Force moves an object through a Displacement in the same direction as the force. Work is a scalar quantity. In physics, in order to get the Work done, there must be a force and there must be motion caused by that force. If the force F is constant, and the displacement d is exactly in the same direction as the direction of the force, then the magnitude of the work is easy to calculate: W = F d constant F has the same direction as d (6.2) Work if the force is not in the same direction as the displacement If the constant force F is not in the same direction as the displacement vector d, then the magnitude of the work done is given by W = F d cos θ constant F separated by angle θ from d (6.1) W = F d Work is the dot product of the vectors F and d (6.20) Kinetic Energy The Energy of Moving Objects Consider the following logical progression: When a force acts on an object of mass m for a specific period of time causing the object to move, then work W is done by the force. If the object has a force acting on it, then it will accelerate. If the object was initially at rest, then after the force stops acting the object will have a speed v. We can define the Kinetic Energy K of the object to be K = 1 2 mv2 (VERY IMPORTANT EQUATION) (6.18) The kinetic energy of an object moving with speed v is equal to the work done by the force which made the object move from rest to the speed v. In physics Power is defined as the rate at which work is done by a force. For a constant (time-independent) force F, one has the following rate of work for a changing displacement vector s P = dw dt = d dt ( F s) = F d s dt = F v (6.27)
Lecture 9: Kinetic Energy and Work 2 Simple Example of WORK in One Dimension The rule to remember about work is: If it moves where you push it, then you re getting the work done. Consider an example of a box resting on a frictionless table. A horizontal force F is applied to that box. By Newton s second law, the box will acquire an acceleration. After the box moves a displacement d along the table, the force d is removed. The work done in this case is given by W = F d constant F has the same direction as d (6.2) Units of Work In the standard SI system of units, work has the units of Newtons times Meters, so one could express the units of work as N m. There is a special name for the Newton meter, and this name is the Joule (J) after the English physicist who did early research into forces and energy. In the cgs system of units (centimeter gram second), work has the units of dynes times cm, and the dyne cm is given the special name erg. There are 10 7 ergs in one joule of energy. There is also the kilowatt-hour (what you get charged for in you electric bill) which is energy NOT power. A kw-hr = 3.6 x 10 6 J. Finally, there is the atomic, nuclear, and high energy physics units called the electron-volt abbreviated as ev, the million-electron-volt unit MeV, and lastly the giga-electron-volt unit GeV = 10 9 ev. An ev = 1.6 x 10 19 J.
Lecture 9: Kinetic Energy and Work 3 Work in Two (and even Three) Dimensions The next simplest case of Work is for two dimensions when there is a constant Force but the displacement vector is not in the same direction as the force vector. Consider again an object resting on a frictionless table. In this case the applied force is not in the horizontal direction, but at some angle Θ to the horizontal. Also, the vertical component of the applied force (= F sin Θ) is not sufficient to raise the object off the table, but since there is no friction the horizontal component of the force (= F cos Θ) causes the object to accelerate along the horizontal direction. In this case the work is done by just the horizontal component of the force. It is that component of the force which moves the object in the same direction as the component itself. W = (F cos Θ)s = F d cos Θ constant F separated by angle Θ from d (6.1) W = F d Work is the dot product of the vectors F and d (6.4) The DOT PRODUCT of two vectors The dot product is a means of multiplying two vectors quantities in order to produce a scalar quantity. The dot product is equal to the product of the magnitudes of the two vectors times the cosine of the angle between the two vectors (with F = F x î+ F y ĵ+ F z ˆk and d = dx î+ d y ĵ+ d z ˆk) W = F d = F d cos Θ = F x d x + F y d y + F z d z If instead of a finite displacement vector d one has a series of infinitesimal displacement vectors dr = dx î + dy ĵ + dz ˆk, then there is an infinitesimal work element given by: = W = 2 dw = F dr = F x dx + F y dy + F z dz (6.10) dw = r2 F dr x2 = 1 r 1 x 1 F x dx + y2 y 1 F y dy + z2 z 1 F z dz
Lecture 9: Kinetic Energy and Work 4 Example of Work in Two Dimensions A box is dragged across a rough floor by a constant force of magnitude 50 N. The force makes an angle of 37 o with the horizontal. A frictional force of 10 N retards the motion. The box is moved a total of 3 m along the floor. Calculate the work done by all the forces acting on the box. First indicate in the free body diagram all the forces acting on the box. Then indicate the displacement vector. You should realize right away that the all forces acting in the vertical direction do no work. The only work is done by those forces which act in, or have a component in, the horizontal direction. Work done by the applied 50 N force W F = (F cos Θ)s = (50 N cos 37 o )(3 m) = 120 N-m = 120 J Work done by the frictional 10 N force W f = (f cos 180 o )(3 m) = ( f)(3 m) = ( 10 N)(3 m) = 30 J Note the negative sign for the work done by the force of friction. The negative sign means that work is done on (or against) the friction force rather than by the frictional force. Forces of friction never produce positive work since there are always opposite to the direction of motion. The net work in this case is the sum of the work of the 50 N force and the work done against friction W net = W F + W f = 120 30 = 90 J
Lecture 9: Kinetic Energy and Work 5 Work and Kinetic Energy Take the same example as before, but now consider that the box has a mass of 5 kg. We define the kinetic energy of the box to be the product K = 1 2 mv2 where v is the speed of the box after it has been moved 3 m. compute v? How can we The speed v after the box has moved 3 m can be computed once the acceleration is known. We would use the kinematics equation v 2 (x) = v 2 0 + 2a(x x 0 ) = v 2 (x = 3) = 2a(3) The box starts from rest (v 0 = 0) and the initial position can be taken as the origin (x 0 = 0). How can we compute the acceleration a? The acceleration a can be computed from Newton s second law if you know the accelerating force and the mass. In this case the accelerating force is along the x direction and is given by F x = F cos Θ f = 50 cos 37 o 10 = 30 N F x = ma Newton s second law = a = F x m = 30 N 5 kg = 6 m/s2 Evaluate the Kinetic Energy The speed may be calculated now v(x = 3) = 2a(3) = 2 6 3 = 6.0 m/s K = 1 2 mv2 = 1 2 5(6)2 = 90 J The kinetic energy of the moving box equals the work done by the forces producing the motion = Work-Kinetic Energy Theorem KINETIC ENERGY = WORK DONE BY A FORCE
Lecture 9: Kinetic Energy and Work 6 Work Done by a Spring A spring is an example of a device which will generate a linearly increasing force as a function of how much the spring is stretched or how much it is compressed. The force generated by a spring when it is stretched or compressed if given by Hooke s Law F spring = kx x displacement of the spring from its unstretched position k force constant of the spring Suppose the spring has been compressed from an initial position x i to a final position x f where x f < x i. The spring opposes this compression and the work done ON the spring is as follows: W = xf x i F spring (x)dx = W = 1 2 kx2 ] x=xi xf x i ( kx)dx = k x=x f = 1 2 k(x2 f x 2 i ) xf x i (x)dx Worked Example A block is lying on a smooth horizontal surface and is connected to a spring with a force constant k equal to 80 N/m. The spring has been compressed 3.0 cm from its equilibrium (= unstretched) position. How much work does the spring do on the block when the block moves from x i = 3.0 to x f = 0.0 cm? W = 1 2 k(x2 f x 2 i ) = 1 2 (80 N/m)(0.030 m)2 = 3.6 x 10 2 J Note that the result is positive, meaning that Work is done BY the spring in this case.
Lecture 9: Kinetic Energy and Work 7 Work, Energy, and Power The last physical quantity introduced in chapter 6 is Power. Power is defined as the rate at which work is done. The more work done in a given amount of time, then the more power that is being produced. Power can either be average power or instantaneous power P = W t W P (t) = lim t 0 t = dw dt The units of power in the SI system is Joules/second which is defined as a watt (W) after James Watt the inventor of the workable steam engine. There is also the kilowatt which is 1,000 watts. Finally electric power bills are usually expressed in terms of kilowatt hours. By the definition of the power unit, you can see that a kilowatt hour is not a unit of power, but rather a unit of energy. What you buy from the electric company is total energy consumption as opposed to rate of energy consumption. It is not how fast you use the electricity at any one time, but rather what is the total amount of electricity which you use Worked Example A weightlifter lifts 250 kg through a vertical distance of 2 m in a time of 1.5 seconds. What is his average power output? First compute the work done, W, and then divide by the time to do that work, t. Here the weightlifter exerts a force equal to the weight of 250 kg, and that force is used through a distance of 2 meters. The average power is then P = W t = W = F d = Mgd = 250 9.8 2 = 4, 900 J 4, 900 J 1.5s = 3, 267 W (greater than 4.3 horsepower) The horsepower unit is a British system unit which is equal to 746 Watts. In principle it is the power output of a standard horse whatever that is.